Chapter 7 Rotational Motion and The Law of Gravity 1
The Radian The radian is a unit of angular measure The radian can be defined as the arc length s along a circle divided by the radius r s θ = r 2
More About Radians Comparing degrees and radians 360 1rad = = 57. 3 2π Converting from degrees to radians π θ [ rad] = θ 180 [degrees] 3
Angular Displacement Axis of rotation is the center of the disk Need a fixed reference line During time t, the reference line moves through angle θ 4
Rigid Body Every point on the object undergoes circular motion about the point O All parts of the object of the body rotate through the same angle during the same time The object is considered to be a rigid body This means that each part of the body is fixed in position relative to all other parts of the body 5
Angular Displacement, cont. The angular displacement is defined as the angle the object rotates through during some time interval Δ = θ θ θ f i The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement 6
Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval ω av θ θ Δθ t t Δt f i = = f i 7
Angular Speed, cont. The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero Units of angular speed are radians/sec rad/s Speed will be positive if θ is increasing (counterclockwise) Speed will be negative if θ is decreasing (clockwise) 8
Average Angular Acceleration The average angular acceleration α of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: α av Δ = f i = t t Δt ω ω ω f i 9
Angular Acceleration, cont Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration 10
Angular Acceleration, final The sign of the acceleration does not have to be the same as the sign of the angular speed The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero 11
Analogies Between equations for Linear and Rotational Motion 12
Relationship Between Angular and Linear (tangential) Quantities Displacement s = θr Tangential speed v = ωr t Tangential acceleration a t = αr Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion 13
QQ4 Andrea and Chuck are riding on a merry-goround. Andrea rides on a horse at the outer rim of the circular platform, twice as far from the center of the circular platform as Chuck, who rides on an inner horse. When the merrygo-round is rotating at a constant angular speed, Andrea s angular speed is : (a) twice Chuck s (b) the same as Chuck s (c) half of Chuck s (d) impossible to determine. 14
QQ5 Andrea and Chuck are riding on a merry-goround. Andrea rides on a horse at the outer rim of the circular platform, twice as far from the center of the circular platform as Chuck, who rides on an inner horse. When the merrygo-round is rotating at a constant angular speed, Andrea s tangential speed is (a) twice Chuck s (b) the same as Chuck s (c) half of Chuck s (d) impossible to determine. 15
Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration The centripetal acceleration is due to the change in the direction of the velocity 16
Centripetal Acceleration, cont. Centripetal refers to centerseeking The direction of the velocity changes The acceleration is directed toward the center of the circle of motion 17
Centripetal Acceleration The magnitude of the centripetal acceleration is given by a c v r 2 The centripetal acceleration can be related to the angular velocity a = C = ω 2 r The direction of centripetal acceleration (and related centripetal force) is toward the center of the circle 18
QQ7 An object moves in a circular path with constant speed v. Which of the following statements is true concerning the object? (a) Its velocity is constant, but its acceleration is changing. (b) Its acceleration is constant, but its velocity is changing. (c) Both its velocity and acceleration are changing. (d) Its velocity and acceleration remain constant. 19
Total Acceleration The tangential component of the acceleration is due to changing speed The centripetal component of the acceleration is due to changing direction Total acceleration can be found from these components 2 a = + a t a 2 C 20
Vector Nature of Angular Quantities Angular displacement, velocity and acceleration are all vector quantities Direction can be more completely defined by using the right hand rule Grasp the axis of rotation with your right hand Wrap your fingers in the direction of rotation Your thumb points in the direction of ω 21
Velocity Directions, Example In a, the disk rotates clockwise, the velocity is into the page In b, the disk rotates counterclockwise, the velocity is out of the page 22
Acceleration Directions If the angular acceleration and the angular velocity are in the same direction, the angular speed will increase with time If the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time 23
Forces Causing Centripetal Acceleration Newton s Second Law says that the centripetal acceleration is accompanied by a force, directed towards the center F C = ma C F C stands for any force that keeps an object following a circular path Tension in a string Gravity Force of friction Normal force (pressure from the support surface) Combination of forces. 24
Problem Solving Strategy Draw a free body diagram, showing and labeling all the forces acting on the object(s) Choose a coordinate system that has one axis perpendicular to the circular path and the other axis tangent to the circular path The normal to the plane of motion is also sometimes needed 25
Problem Solving Strategy, cont. Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, F C ) Use Newton s second law The directions will be radial, normal, and tangential The acceleration in the radial direction will be the centripetal acceleration Solve for the unknown(s) 26
Applications of Forces Causing Centripetal Acceleration Many specific situations will use forces that cause centripetal acceleration Level curves Banked curves Horizontal circles Vertical circles 27
Level Curves Friction is the force that produces the centripetal acceleration Can find the frictional force, µ, or v v = μrg 28
Banked Curves A component of the normal force adds to the frictional force to allow higher speeds tanθ = or a c = 2 v rg g tanθ 29
Vertical Circle Look at the forces at the top of the circle The minimum speed at the top of the circle can be found v top = gr 30
Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. m m F = G 1 r 2 2 31
Universal Gravitation, 2 G is the constant of universal gravitational G = 6.673 x 10-11 N m² /kg² This is an example of an inverse square law 32
Universal Gravitation, 3 The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1 The forces form a Newton s third law actionreaction 33
Universal Gravitation, 4 The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center This is called Gauss Law 34
Gravitation Constant Determined experimentally Henry Cavendish 1798 The light beam and mirror serve to amplify the motion 35
Applications of Universal Gravitation Acceleration due to gravity g will vary with altitude M g = G r E 2 36
Gravitational Potential Energy PE = mgy is valid only near the earth s surface For objects high above the earth s surface, an alternate expression is needed MEm PE = G Zero reference level is infinitely far from the earth r 37
Escape Speed The escape speed is the speed needed for an object to soar off into space and not return 2GME v esc = R E For the earth, v esc is about 11.2 km/s Note, v is independent of the mass of the object 38
Various Escape Speeds The escape speeds for various members of the solar system Escape speed is one factor that determines a planet s atmosphere 39
Kepler s Laws All planets move in elliptical orbits with the Sun at one of the focal points. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals. The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet. 40
Kepler s Laws, cont. Based on observations made by Brahe Newton later demonstrated that these laws were consequences of the gravitational force between any two objects together with Newton s laws of motion 41
Kepler s First Law All planets move in elliptical orbits with the Sun at one focus. Any object bound to another by an inverse square law will move in an elliptical path Second focus is empty 42
Kepler s Second Law A line drawn from the Sun to any planet will sweep out equal areas in equal times Area from A to B and C to D are the same 43
Kepler s Third Law The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet. 2 3 T = Kr For orbit around the Sun, K = K S = 2.97x10-19 s 2 /m 3 K is independent of the mass of the planet 44
Communications Satellite A geosynchronous orbit Remains above the same place on the earth The period of the satellite will be 24 hr r = h + R E Still independent of the mass of the satellite 45